In the figure,   AD = CD and AB = CB . 

(i) Prove that Line BD bisects angle  ADC (i.e. that line BD cuts angle  ADC into two equal angles).

(ii) Prove that Line AC and Line DB are perpendicular.


Maplesnowy  Mar 13, 2018

Since  AD  = CD

Then the angles opposite these sides are also equal

So angle DAC  = angle DCA


And if AB = CB, then the angles opposite thes sides are also equal

So angle CAB  = angle ACB


So angle DAC + angle CAB  = angle DCA + angle ACB

So angle DAB   = angle CDB


Then, by  SAS....triangle  DAB  = triangle DCB


Then angle ADB  = angle CDB.....so angle ADC is bisected


And angle  DAC  = angle DCA


And angle DAC  = angle DCA

So....by ASA....triangle EAD  = triangle   ECD


So angle AED  = angle CED


And, by Euclid.....a line standing upon a line that makes adjacent angles equal means that the lines are perpendicular....so AC  and Db are perpendicular



cool cool cool

CPhill  Mar 13, 2018

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